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Product of products of divisors of divisors of n.
11

%I #27 Apr 09 2023 15:31:26

%S 1,2,3,16,5,216,7,1024,81,1000,11,2985984,13,2744,3375,1048576,17,

%T 34012224,19,64000000,9261,10648,23,63403380965376,625,17576,59049,

%U 481890304,29,19683000000000,31,34359738368,35937,39304,42875,4738381338321616896,37,54872

%N Product of products of divisors of divisors of n.

%C a(n) = Product_{d|n} A007955(d) where A007955(m) = product of divisors of m.

%C From _G. C. Greubel_, Dec 31 2015: (Start)

%C for n>=1: 10^3|a(10*n), 10^6|a(20*n), 10^9|a(30*n).

%C for n>=0: 10^6|a(60*n+50), 10^9|a(60*n+70). (End)

%H G. C. Greubel, <a href="/A266265/b266265.txt">Table of n, a(n) for n = 1..1200</a>

%F a(p) = p for p = prime.

%F a(n) = Product_{d|n} d^tau(n/d). - _Ridouane Oudra_, Apr 09 2023

%e For n = 6; with b(n) = A007955(n); a(6) = b(1)*b(2)*b(3)*b(6) = 1*2*3*36 = 216.

%p A266265 := proc(n)

%p mul( A007955(d),d=numtheory[divisors](n)) ;

%p end proc:

%p seq(A266265(n),n=1..10) ; # _R. J. Mathar_, Feb 13 2019

%t Table[Product[Times @@ Divisors@ d, {d, Divisors@ n}], {n, 38}] (* _Michael De Vlieger_, Dec 31 2015 *)

%o (Magma) [&*[&*[b: b in Divisors(d)]: d in Divisors(n)]: n in [1..100]]

%o (PARI) a(n) = {my(d = divisors(n)); prod(k=1, #d, dd = divisors(d[k]); prod(kk=1,#dd, dd[kk]));} \\ _Michel Marcus_, Dec 27 2015

%Y Cf. A007955 (product of divisors of n), A175317 (sum of products of divisors of divisors of n), A206032 (product of sums of divisors of divisors of n).

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Dec 25 2015